Correlation Time Tau Research Articles

Determination of protein rotational correlation time from NMR relaxation data at various solvent viscosities

An accurate determination of the overall rotation of a protein plays a crucial role in the investigation of its internal motions by NMR. In the present work, an innovative approach to the determination of the protein rotational correlation time tau(R) from the heteronuclear relaxation data is proposed. The approach is based on a joint fit of relaxation data acquired at several viscosities of a protein solution. The method has been tested on computer simulated relaxation data as compared to the traditional tau(R) determination method from T(1)/T(2) ratio. The approach has been applied to ribonuclease barnase from Bacillus amyloliquefaciens dissolved in an aqueous solution and deuterated glycerol as a viscous component. The resulting rotational correlation time of 5.56 +/- 0.01 ns and other rotational diffusion tensor parameters are in good agreement with those determined from T(1)/T(2) ratio.

Triple Quantum Decoherence under Multiple Refocusing: Slow Correlated Chemical Shift Modulations of C′ and N Nuclei in Proteins

A new experiment allows the identification of residues that feature slow conformational exchange in macromolecules. Rotations about dihedral angles that are slower than the global correlation time tau(c) cause a modulation of the isotropic chemical shifts of the nuclei. If these fluctuations are correlated they induce a differential line broadening between three-spin single-quantum and triple-quantum coherences involving three nuclei such as the carbonyl C', the neighbouring amide nitrogen N and the amide proton H(N) belonging to a pair of consecutive amino acids. A cross-correlated relaxation rate R (CS/CS)(C'N) can be determined that corresponds to the sum of the isotropic and anisotropic contributions to the chemical shift modulations of the carbonyl carbon and nitrogen nuclei. Only the isotropic contributions depend on the pulse repetition rate of a multiple-refocusing sequence. An attenuation of the relaxation rate with increasing pulse repetition rate can therefore be attributed to slow motions. The asparagine N25 residue of ubiquitin, located in the first alpha-helix, is shown to feature significant slow conformational exchange.

Effects of cross correlation on the relaxation time of a bistable system driven by cross-correlated noise.

We study the effects of correlations between additive and multiplicative noise on relaxation time in a bistable system driven by cross-correlated noise. Using the projection-operator method, we derived an analytic expression for the relaxation time T(c) of the system, which is the function of additive (alpha) and multiplicative (D) noise intensities, correlation intensity lambda of noise, and correlation time tau of noise. After introducing a noise intensity ratio and a dimensionless parameter R=D/alpha, and then performing numerical computations, we find the following: (i) For the case of R<1, the relaxation time T(c) increases as R increases. (ii) For the cases of R>/=1, there is a one-peak structure on the T(c)-R plot and the effects of cross-correlated noise on the relaxation time are very notable. (iii) For the case of R<1, T(c) almost does not change with both lambda and tau, and for the cases of R>/=1, T(c) decreases as lambda increases, however T(c) increases as tau increases. lambda and tau play opposite roles in T(c), i.e., lambda enhances the fluctuation decay of dynamical variable and tau slows down the fluctuation decay of dynamical variable.

Mean escape time over a fluctuating barrier.

An approximate method for studying activation over a fluctuating barrier of potential is proposed. It involves considering separately the slow and fast components of barrier fluctuations, and it applies for any value of their correlation time tau. It gives exact results for the limiting values tau-->0 and tau--> infinity, and the agreement with numerics in between is also excellent, both for dichotomic and Gaussian barrier perturbations.

Reexamination of the evolution of the dynamic susceptibility of the glass former glycerol

The dielectric data of glycerol compiled by Lunkenheimer et al. [Contemp. Phys. 41, 15 (2000)] are reanalyzed within a phenomenological approach on the one hand, and within mode coupling theory (MCT), on the other. We present a complete interpolation of the dielectric data covering 17 decades in frequencies. The crossover temperature extracted from the phenomenological analysis of the slow response at low temperatures and defined by the emergence of the excess wing upon cooling agrees well with the critical temperature extracted from a MCT analysis of the dynamics at high temperatures including data that were not used in the first MCT analysis of glycerol by Lunkenheimer et al. [Phys. Rev. Lett. 77, 318 (1996)]. The crossover temperature is found to be T(c)=288+/-3 K, which is significantly higher than previously reported. Extracting the nonergodicity parameter f, the characteristic anomaly is only found when 1-f is inspected, since f is very close to 1. No difference for the evolution of the dynamic susceptibility is observed for the nonfragile system glycerol with respect to fragile glass formers provided that the evolution of the dynamics is studied as a function of the correlation time tau(alpha).

Stochastic resonance driven by two different kinds of colored noise in a bistable system.

The phenomenon of stochastic resonance in a bistable nonlinear system is investigated when both the multiplicative noise and the coupling between additive and multiplicative noise are colored with different values of noise correlation time tau(1) and tau(2). Combining the functional analysis and unified colored noise approximation, the two different kinds of colored noise in the nonlinear system can be simplified. The signal-to-noise ratio is calculated when a weakly periodic signal is added to the system. It is found that there appears a transition between one peak and two peaks in the curve of the signal-to-noise ratio when either the noise correlation time tau(1) and tau(2) or the coupling strength lambda between additive and multiplicative noise is increased. The transition between one and two peaks depending on tau(1) and lambda is more complex than that depending on tau(2).

Relaxation, equilibrium oligomerization, and molecular symmetry of the VASP (336-380) EVH2 tetramer.

An investigation of the structural and dynamic properties of the C-terminal fragment of the human protein VASP (VASP 336-380) has been performed. Full length VASP has been shown to be tetrameric in solution, and the C-terminal 45 residues of the protein have been suggested to be responsible for the oligomerization. We have expressed and purified a C-terminal fragment of the human VASP protein from residue 336-380. It was found to form a stable domain in its own right. The fragment was shown by CD spectroscopy to form a helical structure, stable under a wide range of temperature and pH conditions. A (15)N-HSQC-experiment exhibits only one set of peaks, suggesting a high degree of symmetry for a putative oligomer. Measurements of the rotational correlation time tau(C) of the molecule and analytical ultracentrifugation data show VASP (336-380) to be entirely tetrameric in solution. The secondary structure was confirmed from a (15)N-NOESY-HSQC experiment and is completely alpha-helical. We conclude that VASP (336-380) forms a tetramer in solution via a coiled coil arrangement and is solely responsible for tetramerization of full-length VASP.

Domain flexibility in ligand-free and inhibitor-bound Escherichia coli adenylate kinase based on a mode-coupling analysis of 15N spin relaxation.

Adenylate kinase from Escherichia coli (AKeco), consisting of a 23.6-kDa polypeptide chain folded into domains CORE, AMPbd, and LID catalyzes the reaction AMP + ATP <--> 2ADP. The domains AMPbd and LID execute large-amplitude movements during catalysis. Backbone dynamics of ligand-free and AP(5)A-inhibitor-bound AKeco is studied with slowly relaxing local structure (SRLS) (15)N relaxation, an approach particularly suited when the global (tau(m)) and the local (tau) motions are likely to be coupled. For AKeco tau(m) = 15.1 ns, whereas for AKeco*AP(5)A tau(m) = 11.6 ns. The CORE domain of AKeco features an average squared order parameter, <S(2)>, of 0.84 and correlation times tau(f) = 5-130 ps. Most of the AKeco*AP(5)A backbone features <S(2)> = 0.90 and tau(f) = 33-193 ps. These data are indicative of relative rigidity. Domains AMPbd and LID of AKeco, and loops beta(1)/alpha(1), alpha(2)/alpha(3), alpha(4)/beta(3), alpha(5)/beta(4), and beta(8)/alpha(7) of AKeco*AP(5)A, feature a novel type of protein flexibility consisting of nanosecond peptide plane reorientation about the C(i-1)(alpha)-C(i)(alpha) axis, with correlation time tau(perpendicular) = 5.6-11.3 ns. The other microdynamic parameters underlying this dynamic model include S(2) = 0.13-0.5, tau(parallel) on the ps time scale, and a diffusion tilt beta(MD) ranging from 12 to 21 degrees. For the ligand-free enzyme the tau(perpendicular) mode was shown to represent segmental domain motion, accompanied by conformational exchange contributions R(ex) < or = 4.4 s(-1). Loop alpha(4)/beta(3) and alpha(5)/beta(4) dynamics in AKeco*AP(5)A is related to the "energetic counter-balancing of substrate binding" effect apparently driving kinase catalysis. The other flexible AKeco*AP(5)A loops may relate to domain motion toward product release.

The impact of rigidity and water exchange on the relaxivity of a dendritic MRI contrast agent.

Variable-temperature, multiple magnetic field (17)O NMR, EPR and variable-temperature (1)H nuclear magnetic relaxation dispersion (NMRD) measurement techniques have been applied to Gadomer 17, a new dendritic contrast agent for magnetic resonance imaging. The macromolecule bears 24 Gd(dota)-monoamide chelates (dota=N,N',N",N"'-tetracarboxymethyl-1,4,7,10-tetraazacyclododecane) attached to a lysine-based dendrimer. (17)O NMR and (1)H NMRD data were analysed simultaneously by incorporating the Lipari-Szabó approach for the description of rotational dynamics. The water exchange rate k(298)(ex)was found to be (1.0 +/- 0.1) x 10(6) s(-1), a value similar to those measured for other Gd(dota)-monoamide complexes, and the activation parameters DeltaH++ =24.7 +/- 1.3 kJ mol(-1) and DeltaS++ = -47.4 +/- 0.2 JK(-1) mol(-1). The internal flexibility of the macromolecule is characterised by the Lipari-Szabó order parameter S(2)=0.5 and a local rotational correlation time tau(298)(l)= 760 ps, whereas the global rotational correlation time of the dendrimer is much longer, tau(298)(g)=3050 ps. The analysis of proton relaxivities reveals that, beside slow water exchange, internal flexibility is an important limiting factor for imaging magnetic fields. Electronic relaxation, though faster than in similar, but monomeric, Gd(III) chelates, does not limit proton relaxivity of this contrast agent (r(1)=16.5mM(-1)s(-1) at 298 K and 20 MHz). This analysis provides direct clues for the design of high-efficiency contrast agents.

Backbone dynamics of the cytotoxic ribonuclease alpha-sarcin by 15N NMR relaxation methods.

The cytotoxic ribonuclease alpha-sarcin is a 150-residue protein that inactivates ribosomes by selectively cleaving a single phosphodiester bond in a strictly conserved rRNA loop. In order to gain insights on the molecular basis of its highly specific activity, we have previously determined its solution structure and studied its electrostatics properties. Here, we complement those studies by analysing the backbone dynamics of alpha-sarcin through measurement of longitudinal relaxation rates R1, off resonance rotating frame relaxation rates R1 rho, and the 15N[1H] NOE of the backbone amide 15N nuclei at two different magnetic field strengths (11.7 and 17.6 T). The two sets of relaxation parameters have been analysed in terms of the reduced spectral density mapping formalism, as well as by the model-free approach. alpha-Sarcin behaves as an axial symmetric rotor of the prolate type (D(axially)/D(radially)=1.16 +/- 0.02) which tumbles with a correlation time tau(m) of 7.54 +/- 0.02 ns. The rotational diffusion properties have been also independently evaluated by hydrodynamic calculations and are in good agreement with the experimental results. The analysis of the internal dynamics reveals that alpha-sarcin is composed of a rigid hydrophobic core and some exposed segments which undergo fast (ps to ns) internal motions. Slower motions in the mu s to ms time scale are less abundant and in some cases can be assigned to specific motional processes. All dynamic data are discussed in relation to the role of some particular residues of alpha-sarcin in the process of recognition of its ribosomal target.

Oligomerization of paramagnetic substrates result in signal amplification and can be used for MR imaging of molecular targets.

Magnetic resonance imaging (MRI) has evolved into a sophisticated, noninvasive imaging modality capable of high-resolution anatomical and functional characterization of transgenic animals. To expand the capabilities MRI, we have developed a novel MR signal amplification (MRamp) strategy based on enzyme-mediated polymerization of paramagnetic substrates into oligomers of higher magnetic relaxtivity. The substrates consist of chelated gadolinium covalently bound to phenols, which then serve as electron donors during enzymatic hydrogen peroxide reduction by peroxidase. The converted monomers undergo rapid condensation into paramagnetic oligomers leading to a threefold increase in atomic relaxtivity (R1/Gd). The observed relaxtivity changes are largely due to an increase in the rotational correlation time tau r of the lanthanide. Three applications of the developed system are demonstrated: (1) imaging of nanomolar amounts of an oxidoreductase (peroxidase); (2) detection of a model ligand using an enzyme-linked immunoadsorbent assay format; and (3) imaging of E-selectin on the surface of endothelial cells probed for with an anti-E-selectin-peroxidase conjugate. The development of "enzyme sensing" probes is expected to have utility for a number of applications including in vivo detection of specific molecular targets. One particular advantage of the MRamp technique is that the same paramagnetic substrate can be potentially used to identify different molecular targets by attaching enzymes to various antibodies or other target-seeking molecules.

Oligomerization of Paramagnetic Substrates Result in Signal Amplification and can be Used for MR Imaging of Molecular Targets

Magnetic resonance imaging (MRI) has evolved into a sophisticated, noninvasive imaging modality capable of high-resolution anatomical and functional characterization of transgenic animals. To expand the capabilities MRI, we have developed a novel MR signal amplification (MRamp) strategy based on enzyme-mediated polymerization of paramagnetic substrates into oligomers of higher magnetic relaxtivity. The substrates consist of chelated gadolinium covalently bound to phenols, which then serve as electron donors during enzymatic hydrogen peroxide reduction by peroxidase. The converted monomers undergo rapid condensation into paramagnetic oligomers leading to a threefold increase in atomic relaxtivity (R1/Gd). The observed relaxtivity changes are largely due to an increase in the rotational correlation time tau r of the lanthanide. Three applications of the developed system are demonstrated: (1) imaging of nanomolar amounts of an oxidoreductase (peroxidase); (2) detection of a model ligand using an enzyme-linked immunoadsorbent assay format; and (3) imaging of E-selectin on the surface of endothelial cells probed for with an anti-E-selectin-peroxidase conjugate. The development of "enzyme sensing" probes is expected to have utility for a number of applications including in vivo detection of specific molecular targets. One particular advantage of the MRamp technique is that the same paramagnetic substrate can be potentially used to identify different molecular targets by attaching enzymes to various antibodies or other target-seeking molecules.

Dielectric and elastic properties of liquid crystals.

The structural properties, the static and relaxation dielectric coefficients [epsilon(j) and epsilon(j)(omega) (j= ||, perpendicular)], the rotational diffusion constants D( perpendicular) and D( ||), the orientational correlation times tau(1)(i0) (i=0,1), and the bulk elastic constants K(i) (i=1,2,3) are investigated for polar liquid crystals, such as 4-n-pentyl-4(')-cyanobiphenyl (5CB). epsilon(j) are calculated by a combination of the existing molecular theory and statistical-mechanical approach (SMA) that takes into account translational and orientational correlations as well as their coupling, whereas epsilon(j)(omega) are calculated by combining SMA and nuclear magnetic resonance relaxation theory, both based on a rotational diffusion model in which the reorientation of an individual molecule is assumed as stochastic Brownian motion in a potential of mean torque. Reasonable agreement between the calculated and experimental values of epsilon(j) and epsilon(j)(omega) for 5CB is obtained. The bulk Frank elastic constants K(i) (i=1,2,3), for splay, twist, and bend distortion modes, as well as their ratios K(3)/K(1) and K(2)/K(1) are also obtained.

Analytical solution to the Lipari-Szabo model based on the reduced spectral density approximation offers a novel protocol for extracting motional parameters.

An analytical solution to the Lipari-Szabo model is derived for isotropic overall tumbling. The parameters of the original Lipari-Szabo model, the order parameter S2 and the effective internal correlation time tau(e), are calculated from two values of the spectral density function. If additionally the spectral density value J(0) is known, the exchange contribution R(ex) term can also be determined. The overall tumbling time tau(c) must be determined in advance, for example, from T1/T2 ratios. The required spectral density values are obtained by reduced spectral density mapping from T1, T2, and NOE measurements. Our computer simulations show that the reduced spectral density mapping is a very good approximation in almost all cases in which the Lipari-Szabo model is applicable. The robustness of the analytical formula to experimental errors is also investigated by extensive computer simulations and is found to be similar to that of the fitting procedures. The derived formulas were applied to the experimental 15N relaxation data of ubiquitin. Our results agree well with the published parameter values of S2 and tau(e), which were obtained from standard fitting procedures. The analytical approach to extract parameters of molecular motions may be more robust than standard analyses and provides a safeguard against spurious fitting results, especially for determining the exchange contribution R(ex).

Viscous flow and jump dynamics in molecular supercooled liquids. II. Rotations.

The rotational dynamics of a supercooled model liquid of rigid A-B dumbbells interacting via a Lennard-Jones potential is investigated along one single isobar. The time-temperature superposition principle, one key prediction of mode-coupling theory (MCT), was studied for the orientational correlation functions C(l). In agreement with previous studies we found that the scaling of C(l) in a narrow region at long times is better at high-l values. However, on a wider time interval the scaling works fairly better at low-l values. Consistently, we observed the remarkable temperature dependence of the rotational correlation time tau(1) as a power law in T-T(c) over more than three orders of magnitude and the increasing deviations from that law on increasing l (T(c) is the MCT critical temperature). For 0.7<T<2, good agreement with the diffusion model is found. For lower temperatures the agreement becomes poorer, and the results are also only partially accounted for by the jump-rotation model. The angular Van Hove function shows that in this region a meaningful fraction of the sample reorientates by jumps of about 180 degrees. The distribution of the waiting times in the angular sites cuts exponentially at long times. At lower temperatures it decays at short times as t(xi-1), with xi=0.34+/-0.04 at T=0.5, in analogy with the translational case. The breakdown of the Debye-Stokes-Einstein relation is observed at lower temperatures, where the rotational correlation times diverge more weakly than the viscosity.

Effects of colored noise on stochastic resonance in a bistable system subject to multiplicative and additive noise.

The effects of colored noise on stochastic resonance (SR) in a bistable system driven by multiplicative colored noise and additive white noise and a periodic signal are studied by using the unified colored noise approximation and the theory of signal-to-noise ratio (SNR) in the adiabatic limit. In the case of no correlations between noises, there is an optimal noise intensities ratio R at which SNR is a maximum that identifies the characteristics of the SR when the correlation time tau of the multiplicative colored noise is small. However, when tau is increased, a second optimal value of R appears, and two peaks appear in the SNR simultaneously. In the case of correlations between noises, the SNR is not only dependent on the correlation time tau, but also on the intensity of correlations between noises. Moreover, the double peak phenomenon can also appear as tau is increased in certain situations.

Rotational viscosity, dynamic phenomena, and dielectric properties in a long-chain liquid crystal: NMR study and theoretical treatment.

The rotational diffusion constants D(perpendicular) and D(parallel), rotational viscosity coefficients gamma(i) (i=1,2), the orientational correlation times tau(L)mn, and the dielectric permittivities for nematic liquid crystals (NLCs) are investigated. gamma(i) are calculated by a combination of existing statistical-mechanical approach (SMA) and NMR relaxation theory, both based on a rotational diffusion model. In the rotational diffusion model, it is assumed that the reorientation of an individual molecule is a stochastic Brownian motion in a certain potential of mean torque. According to the SMA, gamma(i) are found to be a function of temperature, density, rotational diffusion constant for tumbling motions, and the orientational order parameters. The order parameters and rotational diffusion constant are obtained from an analysis of NMR measurements. Reasonable agreement between the calculated and experimental values of gamma(i) for 4-n-octyloxy-4'-cyanobiphenyl (8OCB) is obtained. The orientational correlation times, and the longitudinal and transverse components of the real chi'(j)(omega) and imaginary chi"(j)(omega) (j= parallel, perpendicular) parts of the complex susceptibility tensor for 8OCB molecules in the nematic phase are also obtained.

Calix[4]arenes as selective extracting agents. An NMR dynamic and conformational investigation of the lanthanide(III) and thorium(IV) complexes.

The lanthanide and Th4+ complexes with calix[4]arene ligands substituted either on the narrow or at the wide rim by four coordinating groups behave totally differently as shown by an NMR investigation of the dia- and paramagnetic complexes. Solutions of complexes were prepared by reacting anhydrous metal perchlorate salts with the ligands in dry acetonitrile (CAUTION). Relaxation time T1 titrations of acetonitrile solutions of Gd3+ by calixarenes indicate that ligands subsituted on the narrow rim form stable 1:1 complexes whether they feature four amide groups (1) or four phosphine oxide functions. In contrast, a ligand substituted by four (carbamoylmethyl)-diphenylphosphine oxide moieties on the wide rim (3) and its derivatives from polymeric species even at a 1:1 ligand/metal concentration ratio. Nuclear magnetic relaxation dispersion (NMRD) curves (relaxation rates 1/T1 vs magnetic field strength) of Gd3+, Gd3+.1 and Gd3+.3 perchlorates in acetonitrile are analyzed by an extended version of the Solomon-Bloembergen-Morgan equations. A comparison of the calculated rotational correlation times tau r shows that ligand 3 forms oligomeric Gd3+ species. The chelates of ligand 1 are axially symmetric (C4 symmetry), and the paramagnetic shifts induced by the Yb3+ ion are accounted for quantitatively. The addition of water or of nitrate ions does not modify the geometry of the complex. The metal chelates of 3 and its derivatives adopt a C2 symmetry, and the paramagnetic shifts are interpreted on a semiquantitative basis only. Water and NO3- ions completely labilize the complexes of the heavy lanthanides. The very high selectivity of ligand 3 through the lanthanide series stems from a complex interplay of factors.

The solution structure of [d(CGC)r(aaa)d(TTTGCG)](2): hybrid junctions flanked by DNA duplexes.

The solution structure and hydration of the chimeric duplex [d(CGC)r(aaa)d(TTTGCG)](2), in which the central hybrid segment is flanked by DNA duplexes at both ends, was determined using two-dimensional NMR, simulated annealing and restrained molecular dynamics. The solution structure of this chimeric duplex differs from the previously determined X-ray structure of the analogous B-DNA duplex [d(CGCAAATTTGCG)](2)as well as NMR structure of the analogous A-RNA duplex [r(cgcaaauuugcg)](2). Long-lived water molecules with correlation time tau(c)longer than 0.3 ns were found close to the RNA adenine H2 and H1' protons in the hybrid segment. A possible long-lived water molecule was also detected close to the methyl group of 7T in the RNA-DNA junction but not with the other two thymines (8T and 9T). This result correlates with the structural studies that only DNA residue 7T in the RNA-DNA junction adopts an O4'-endo sugar conformation, while the other DNA residues including 3C in the DNA-RNA junction, adopt C1'-exo or C2'-endo conformations. The exchange rates for RNA C2'-OH were found to be approximately 5-20 s(-1). This slow exchange rate may be due to the narrow minor groove width of [d(CGC)r(aaa)d(TTTGCG)](2), which may trap the water molecules and restrict the dynamic motion of hydroxyl protons. The minor groove width of [d(CGC)r(aaa)d(TTTGCG)](2)is wider than its B-DNA analog but narrower than that of the A-RNA analog. It was further confirmed by its titration with the minor groove binding drug distamycin. A possible 2:1 binding mode was found by the titration experiments, suggesting that this chimeric duplex contains a wider minor groove than its B-DNA analog but still narrow enough to hold two distamycin molecules. These distinct structural features and hydration patterns of this chimeric duplex provide a molecular basis for further understanding the structure and recognition of DNA. RNA hybrid and chimeric duplexes.

Resonances while surmounting a fluctuating barrier

Electronic analog experiments on escape over a fluctuating potential barrier are performed for the case when the fluctuations are caused by Ornstein-Uhlenbeck noise (OUN). In its dependence on the relation between the two OUN parameters (the correlation time tau and noise strength Q) the nonmonotonic variation of the mean escape time T as a function of tau can exhibit either a minimum (resonant activation), or a maximum (inhibition of activation), or both these effects. The possible resonant nature of these features is discussed. We claim that T is not a good quantity to describe the resonancelike character of the problem. Independently of the specific relation between the OUN parameters, the resonance manifests itself as a maximal lowering of the potential barrier during the escape event, and it appears for tau of the order of the relaxation time toward the metastable state.